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CROSBY LOCKWOOD & SON'SLIST OF WORKS

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LONDON :7, STATIONERS' HALL COURT, LUDQATE HILL, E.G.,AND

5, Broadway, Westminster, 8.W.I9I5-


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LJZ

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AIR-SCREWSAn Introduction to the Aerofoil Theory ofScrew Propulsion

BYM. A. S. RIACH

) i

ASSOCIATE FELLOW OF THE AERONAUTICAL SOCIETY

** In mediis qure rigore omni vacant resistantiiS corporumstint in duplicata ratione velocitatum. Newton.

NEW YORKD. APPLETON AND COMPANY

MCMXVI


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PRINTED IN CHEAT liRITAIN


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PREFACEWITH the coming of the Aeroplane the quantitative study ofscrews working in air has assumed a great importance.Formerly in the design of screw propellers for marine workexperiments with models were carried out and the performanceof the full size screw calculated from them, and it was notuntil 1882 that Drzewiecki first drew attention to a possiblemore powerful method of design obtained by considering eachelement along the blade as independent and behaving in thesame manner as if moving through the fluid in a straight line.

This method has since assumed great importance in thepractical design of air-screw blades, and the results obtainedseemed to justify the utilization of this theory as at leastapproximately correct provided certain limits are not exceeded.

In the present work the theory has been assumed to beabsolutely correct, and the results obtained have been carriedto their logical conclusions. This has been done for variousreasons.

It does not make for completeness in any argument if theresults of the initial hypothesis are not carried to their ultimatelogical conclusions, and although in the present instance theresults so obtained may not be completely borne out in practice,yet, in giving an insight into the applications of the theory,and in establishing at any rate an approximate method fordealing with the many cases arising out of the performancesof aircraft, the conclusions arrived at will, it is hoped, not bewithout interest.

In any case the practical application of some of the moreextreme results should not be made without due caution, and


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iv PREFACEin fact they are to be regarded as of an extremely tentativenature. This caution is necessary, for There is no morecommon error than to assume that, because prolonged andaccurate mathematical calculations have been made, the appli-cation of. the result to some fact of nature is absolutely certain.The conclusion of no argument can be more certain than theassumption from which it starts (Whitehead, Introductionto Mathematics ). Mathematics are too often apt to beregarded as capable of creating results, quite independentlyof any initial hypotheses, when they are nothing more than avery useful tool.

I have endeavoured to present the subject of air-screwdesign in as simple a manner as possible, so that the ordinarynon-mathematical reader may be able to follow the train ofreasoning, at any rate as far as its qualitative nature isconcerned.

It may be that the first chapter is unnecessarily drawn out,but it appears to me that in any investigation of this kindthe first essential is to be able to clearly ''visualise what isbeing done, the mere application of analytical processes beingbut a secondary matter.

I have introduced graphical methods wherever it seemed tobe necessary, or where it was impossible to obtain solutionswithout them. At the same time the results for the design ofan air-screw to fulfil any specified outside conditions havebeen put into such a form that it is hoped that the designwill be able to be correctly carried out by the rules given,even if the analytical processes have not been able to befollowed by the reader.

With regard to the possible errors involved in the applica-tion of the results given in the text, these should not be foundto materially affect any but the last chapters of the book.The chapters on

static thrust, efficiency of air-screws from(V) equal to zero up to the velocity of flight, and on directlifting systems, are admittedly of a speculative chara'cter.


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PREFACE vIt did not seem that a work of this kind could be regarded

as complete without some reference to the stresses occurring inan air-screw blade, and accordingly a chapter on centrifugaland bending stresses has been included.

It is hoped that the book will be found to be not withoutinterest to engineers desiring an introduction to the theory ofair-screws, while at the same time it may perhaps conduce toa more scientific study of the subject, in place of what hasbeen aptly described as the make it 4 by 2 methods sodear to the heart of the practical man.

I take this opportunity of thanking Mr. H. Bolas, of theAir Department, Admiralty, for Ids valuable criticisms uponthe proofs. My thanks are also due to Mr. T. E. Ritchie forhis help in reading through the proofs, and to Mr. A. King forhis assistance in the compiling of the various diagrams.The Controller of His Majesty's Stationery Office haspermitted me to quote from certain of the Technical Reports ofthe Advisory Committee for Aeronautics.

M. A. S. R.HENDON.


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CONTENTSINTRODUCTION. PAGE

PRESSURE ON AEROFOILS . ... 1CHAPTER I.

THE PITCH OF AN AIR-SCREW ....... 6CHAPTER II.

THE FORCES ACTING ON AN AIR-SCREW BLADE . . . .14CHAPTER III.

BLADE SHAPE AND EFFICIENCY ....... 25CHAPTER IV.BLADE SECTIONS, AND WORKING FORMULAE . . . . .47

CHAPTER V. LAYING OUT THE AIR-SCREW . . . . . . .72

CHAPTER VI.STRESSES IN AIR-SCREW BLADES ... .77

CHAPTER VII.STATIC THRUST ....


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viii CONTENTSCHAPTER VIII. PAGE

EFFICIENCY OF AN AIR-SCREW AT DIFFERENT SPEEDS OF TRANSLATION 91

CHAPTER IX.DIRECT LIFTING SYSTEMS ........ 109

APPENDIX I.NOTE ON THE INFLUENCE OF ASPECT RATIO .... 1'zi

APPENDIX II.NOTE ON THE EFFECT OF THE INDRAUGHT IN FRONT OF AN

AlR-SCREW . . 123

INDEX . . .125


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AIE-SGEEWSINTRODUCTION.

PKESSURE ON AEROFOILS.THE problem of determining the form of air flow generated byan aerofoil, moving through still air, is one that has so fardefied mathematical analysis. This problem, which appliesequally to water and other fluids, is one for which a solutionhas only been obtained when the aerofoil is a flat plate ofinfinite span.* The analytical results however do not conformwith those obtained by experiment, and it would seem that thestill more complex problem of the curved surface is beyondthe reach of present-day analysis.

The mathematical theory makes the pressure on a flatsurface vary as the square of the general velocity of the stream,and experiment has shown that between fairly large limits invelocity this holds good.

Briefly, if an aerofoil be exposed to a moving current of airor, what is the same thing, if an aeroplane wing be movingthrough still air, the resultant air pressure on the aerofoil maybe expressed by the formula (R = &.S.V2), where (S) denotesthe area of the aerofoil surface, and (V) the velocity of flightof the aerofoil relative to the air. (k) is a constant thenumerical value of which depends upon the units employed inthe measurements of the quantities (R), (S), and (V).

* A further development has recently been obtained by ProfessorG. H. Bryan and Mr. E. Jones. Discontinuous Fluid Motion past aBent Plane, with Special Eeference to Aeroplane Problems. By G. H.Bryan, Sc.D., F.E.S., and Eobert Jones, M.A. (Proceedings of the EoyalSociety, Vol. 91, No. A 630).

B


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2, : : . : AIE-SCEEWSIt can easily be seen that (k) has thus the dimensions of

density. If we introduce the density of the air into theequation, we can write (E = c./o.S.V2 ), where (p) denotesatmospheric density, and the constant (c) is then non-dimensional.

The resultant pressure (E) is composed of two quantities,the pressure on the under surface of the aerofoil and thenegative pressure on the top surface of the aerofoil. Thislatter as a rule forms the principal portion of (E) and in manycases is from three to four times as large as the pressure on theunder surface.

(E) can be split up into two components, measured normalto, and tangential with, the line of flight of the aerofoil.These two components are termed the lift and drag componentsrespectively, and the value of their ratio is termed the lift-drag ratio of the aerofoil.Since (c) has been shown to be non-dimensional, the twoconstants in the expressions for the lift and drag will also benon-dimensional, and accordingly we may write

L = c^.p.S.V 2 ,D = cx.p.S.V2 ,where the suffixes (y) and (x) affixed to (c) denote vertical andhorizontal components respectively.

(cy) and (cx) are known as the absolute lift coefficient andabsolute drag coefficient respectively.

From the above it can be seen that the ratio ^- is equalto the ratio , and so, when estimating the value of the lift-cxdrag ratio of an aerofoil, it is unnecessary to know the value ofthe actual pressures concerned; it is sufficient to know the valueof each of the coefficients.

f*The value of this ratio - is of great importance in aeroplanecxdesign, and the suitability of an aerofoil shape as a wing sectiondepends largely upon this value.

It is always desirable to make this ratio as large as possible.


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INTRODUCTION 3This also applies to the theory of air-screw design based on theaerofoil analogy.

In records of experimental results, the reciprocal of thisratio is sometimes used.nThe ratio -^ can also be expressed as cot 7, where 7 denotescx

the gliding angle of the aerofoil.The numerical values of (cy) and (cx) are found to vary withvariations in the angle of the chord incidence of the aerofoilwith the direction of motion, and their values are usually givenover a large range of angles.

For many forms of aerofoil, the angle corresponding withthe highest value of is found to be in the neighbourhood of 4.cx

The value of (c) for normal incidence, e.g. broad sideon, of a flat surface of infinite span is (*44) on the modernmathematical theory, the actual value found from experimentbeing ( 64).

There is another quantity upon which the values of (cy)and (cx) depend to a certain extent. This quantity is the ratioof the lengths of the span to the chord of an aeroplane wing.It is commonly known as the Aspect Katio.Model aerofoils used for purposes of testing in a windtunnel usually have a value of (6) for this ratio. As a rulethe higher the value of the aspect ratio, the higher thevalue of .

C*It can hardly be said that there is anything essentially new

in the development of a theory of air-screws from the analogypresented with the motion of an aerofoil in a straight line.

This subject was first investigated by Drzewiecki in 1882and has since been developed by Lanchester.*The fundamental hypothesis underlying the whole theoryhere set forth is that :

(1) Each infinitesimal element along the blade of an air-screw may be treated as a separate aerofoil possessing the samecharacteristics as those which an aerofoil, having the same shape

* Aerial Flight, by F. W. Lanchester.


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4 AIR-SCREWSand haviDg an aspect ratio equal to the aspect ratio of thewhole air-screw, would possess ;

(2) The velocity of each blade element, compounded of thetranslational velocity and the circumferential velocity, at thepoint in question, may be treated as causing no appreciablevariation from the V2 law, so that the infinitesimal pressureson every element of the blade may be considered as satisfyingthe relation K = c.p.S.V2 .

Messrs. F. H. Bramwell and A. Fage, of the NationalPhysical Laboratory, say with reference to the application ofthese assumptions : * There are at present two systems mainly employed in thedesign of propellers. The first, which is generally used inthe design of marine propellers, consists in making smallvariations from existing successful designs : it is necessarythat no very great variation should be made at any one time. The second, which has so far been used almost exclusivelyfor the design of aerial propellers, attempts to predict theperformance of a propeller from a consideration of the forceson elementary strips of the blade. This method, if sensiblycorrect, is far more powerful than the older one, as it affordsa means of introducing new features irrespective of whetherthe variation from existing types is small or large. The initial assumptions underlying this method, which hasbeen developed by Lanchester and Drzewiecki, are that theforces on the blades are due directly to the velocities of thevarious sections relative to still air, these velocities being com-pounded of the translational velocity and the circumferentialvelocity at the point in question, and also that the sectionsmay be treated independently. . . . The final conclusion arrived at is that although the methodis not strictly correct, yet in the hands of a careful designer itis probably by far the best method that can be used for thedesign of propellers in the present state of knowledge on thesubject. . . . The question of practical importance, however, is whether

* Technical Eeport of the Advisory Committee for Aeronautics,1912-13.


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INTRODUCTION 5the theory affords a sufficient basis for purposes of design.When examined from this point of view it is founcf that, ifthe range of comparison be limited to that usual in flyingmachines, the experimental and calculated results are insufficiently good agreement, and that so long as tho conditionsunder which the propeller is working are not varied too widely,the theory may be satisfactorily applied. The occasional failureof propellers to satisfy the conditions of design may be due toan overstepping of these limits. In most cases the differencesbetween the calculated and experimental results are notsufficiently large for their effects to be observed in the flyingof the completed aeroplane. . . . On the other hand, it is more probable that most of thediscrepancies are due to the centrifugal forces on the air incontact with the blade ; this must alter the character of theflow round the blade very considerably, and it is perhaps amatter for surprise that the agreement between the calculatedand experimental values is as close as it is, and not that theydo not agree exactly.


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AIE-SCEEWS

CHAPTER I.THE PITCH OF AN AIR-SCREW.

IT is fairly obvious that when an air-screw is moving throughthe aii1 with some definite translational velocity the distance ittraverses in each revolution will be constant, providing theline of flight be horizontal.

FIG. l.

Now if we consider any portion of the blade at a distanceof (x) feet say from the centre of the boss of the air-screw, weshall find that as the air-screw as a whole moves forward, theportion of the blade under consideration moves up someparticular helicoidal path due to the air-screw rotating aboutits axis at the boss centre.


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THE PITCH OF AN AIR-SCREWThe steepness of the helix traversed by the portion of the

blade at radius (x) will depend upon the value of the distancetraversed translationally by the air-screw in each revolution.

It will also depend upon the value of (#), that is upon thedistance of the portion of the blade considered from the centreof the boss of the air-screw.

This may perhaps be more clearly visualized if we imagine

FIG. 2.

a cylinder having a radius of (x) and a depth of (P), as inKg. (1).Or it may be demonstrated by taking a rectangular piece ofpaper and drawing a diagonal line as in Fig. (2).Then if the paper be rolled so as to form a cylinder havingboth ends open, the diagonal line will represent the path orhelix traversed by the point under consideration ; the diameterof the cylinder so formed will represent twice the distance ofthe point considered from the centre of the cylinder (i.e. thecentre of the boss of the air-screw).

The circumference of the base of the cylinder will represent


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8 AIE-SCEEWSthe path that would be traced out, by the point considered, inone revolution, if the air-screw was not moving forward at all,but merely revolving on its axis ; and the length of this circum-ference can be seen to be equal to

(Tr).(the diameter of the cylinder),that is (2.7T.X.), where (x) is the distance of the walls of thecylinder from the centre of the cylinder, corresponding to the

K*)

2TTX

FIG. 3.

distance of the portion of the air-screw blade considered fromits boss centre.

And the depth of the cylinder will then represent thedistance advanced through translationally by the point, andtherefore by the whole air-screw, at each revolution.

If the paper cylinder be now flattened out as in (Fig. 3), itat once becomes apparent that the helix traversed by the pointin each revolution of the air-screw is the hypotenuse of a right-angled triangle, and therefore that its length is equal to

2^ b Euclid L 4 Hr


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10 AIR-SCEEWSthe effective pitch, and the angles formed by the various linesat the base of the triangle represent the helix angles of therespective portions of the blade considered.

We may denote these angles by (A1} A 2 , A3 , ) corre-sponding to radii from the boss centre of (xlt x2 , x3 ).The helix angle of the blade tip, that is the angle at aradius equal to half the diameter of the air-screw, may bedenoted by (0).

It can be seen from the figure that since all the helicoidalpaths of the various portions of the blade meet in a point, theymust all satisfy the relation

tan A = -., p / p \, so that A = tan- 1 ( ^ ),2.7T.X \2.7T.xJgiving the value of the helix angle (A) in degrees for anyradius (x) from the boss centre of the air-screw, providing thevalue of (P) be known.Now suppose that we have an air-screw, and that wemeasure the actual chord angles of the blade to the disc ofrevolution at various distances from the boss centre.

Let us denote these various chord angles by (fa, fa, fa ),corresponding to radii of (xl} x2 , x3 ) from the centre ofthe boss.

In some types of air-screws, these angles are designed sothat they all satisfy the relation

(


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THE PITCH OF AN AIE-SCEEW 11therefore start with the assumption that the chord angles($i 4>z, 3 ) have no specified connection, but that theymay be anything whatever.

Now suppose our air-screw, having the chord angles( $2> $3 ) as defined, to have a translational velocityof (V) feet/sec, and a speed of revolution of (ri) revs. /sec., thenthe distance advanced through in the direction of translationyper revolution is - - feet, and has already been defined as beingthe effective pitch of the air-screw.Suppose that we keep (n) constant, and give to (V) thesuccessive values of (V1} V2 , V3 ).Then the distances advanced through at each revolution

.n , VT V2 V3will be , ,n n nThus it is possible to have an infinite number of values

for the effective pitch of any given air-screw, correspond-ing to an infinite number of values of the translationalvelocity (V).

It is obvious therefore that the effective pitch of any air-screw is not necessarily a fixed quantity, but depends upon thevalues of (V) and (n).

It is usual however to associate the pitch of a screw withthe screw itself, and thus to imagine it to be a fixed quantityfor any given screw.

In order to determine some analogous expression, in thecase of an air-screw, which is a constant quantity for any giventype of air-screw, we may define a particular value of the ratioy- at which there isn

(1) no resultant thrust on the blades in the direction oftranslation ;

(2) no resultant torque on the blades in a directionnormal to the direction of translation, and there-fore tangential to the disc

of revolution of theblades ;

(3) no average reaction on the blades.


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12 AIE-SCKEWSThe three values of 'corresponding to these three cases

will be constants for any given type of air-screw.The quantitative determination of these values of the

effective pitch may be found from the results of the analysisto be proved later.We may denote these values of (P) by (Pj), (P2), and (P3),or by P),P), and (I).\n/i W/2 \n/ 3

The experimental mean pitch of an air-screw has been

FIG. 5.

defined by Mr. F. H. Bramwell as being the value of the ratio -TIfor which there is no thrust on the blades in the direction oftranslation.

This definition corresponds to (1) already given.Suppose then that our air-screw receives a velocity of trans-

lation of (V) feet/sec., and therefore has some definite value ofy, the effective pitch.n


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THE PITCH OF AN AIK-SCREW 13, Denoting the helix angles along the blade at radii of(xlf a?a , a?3 , ) by (Alf A2 , A 3 , ......) and the chordangles by ( lt $2 , 3, ) we may represent the paths ofthe various portions of the blade considered by Fig. (5).We have made the chord angles in each case greater thanthe corresponding helix angles for the sake of clearness.Then it is obvious that, since each of the blade elementsconsidered is moving up the hypotenuse of one of the corre-sponding right-angled triangles, the actual angles at whichthese blade elements move to their respective helicoidal pathsare (


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14 AIK-SCEEWS

CHAPTEE II.THE FORCES ACTING ON AN AIR-SCREW BLADE.

Now it has already been stated that if an aerofoil be movingin a straight line at a velocity of (V), the two components,normal to and tangential with its line of flight, of the resultantpressure exerted upon it by the air can be written

L = cy./>.S.V2 ,D = ^.p.S.V2 ,

and we have the obvious relation already mentioned,L cyD = ;r = cot *

(7) being the gliding angle of the aerofoil at the particularangle of incidence to its line of flight considered.

(7) is the angle which the direction of the resultantair-pressure on an aerofoil makes with the direction of thevertical component of (R), that is the lift.

It will be noticed that the blade sections of an air-screware similar to those used on aeroplane wings, and hence it atonce raises the question: Cannot we treat the sections alongan air-screw blade as if they were aeroplane wings moving atangles of incidence of (a lf a 2 > s> ) ?

It is upon this very assumption that the whole of thetheory of air-screw design is based, as already explained in theIntroduction.

It is also obvious that, in order to be able to correctlyfollow out the consequences of this assumption, we must treateach of the portions of the blade as being of infinitely smalloreadth in the direction of the radius (x).


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FOBCES ACTING ON AN AIE-SOEEW BLADE 15We may then without error sum up the forces on all these

infinitely narrow blade strips, and hence obtain a correctquantitative determination of the characteristics of theair-screw.*

This is of course the ordinary mathematical process ofintegration.

Let us then consider the forces acting upon a strip of blade

FIG. 6.

at a radius of (x) from the boss centre of the air-screw.Fig. (6) shows the two views of the blade, in side elevationand plan.

Since the element of blade is moving up the hypotenuse ofthe right-angled triangle, the two components of the resultant* This is of course an assumption of the theory.


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16 AIE-SCKEWSair-pressure upon it will be normal to and tangential with thehypotenuse of the triangle respectively.

These infinitesimal pressures may be denoted by (dL)and (dD).

Let the width of the blade at radius (x) be denoted by (&).Then, applying the formulae for air-pressure, we have

dL = Cy.p.b.dx.v2,and dD = cx.p.b.dx.v2 .

And the value of (v) is obviously equal tofor, as already shown, the quantity ^P 2 4-4.7r.V is the lengthof the helix traversed by the element of blade at each revolutionof the air-screw, and hence the distance traversed by the elementper second is (n) times this amount, where (n) is equal to thenumber of revolutions of the air-screw per second.

Whence the velocity of the blade element is n. x/PM-^Tr2^2 .So that we havedL = p.ri*.Cy.1>. (P2 + 4.7r2 . 2). dx,dD = p.n'2.cx.b. (P2+ 4.7r2.z2). dx.

Now we are not immediately concerned with (dL), but withits components normal to and tangential with the disc ofrevolution of the blade.

These are(dL. cos A) and (dL. sin A) respectively.

Again, consider the value of (dD), the drag of the element.We have dD = P.n2 .cx.b. (P2+ 4.7i 2.#2). dx,

and this may also be split up into two components normal toand tangential with the disc of revolution of the blade.

These components are

( dD. sin A) and (dD. cos A) respectively.Now the thrust on the element is measured by the com-

ponents of all the forces acting on the element in a direction


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FOBCES ACTING ON AN AIR-SCEEW BLADE 17normal to the disc of revolution of the blade, that is, parallelwith the line of advance of the air-screw.

The forces so acting are seen to be(dL. cos A) and ( dD. sin A),

so that the resultant force on the element in a direction normalto the disc of revolution of the air-screw is (dL. cos A dD. sin A),and this may be denoted by (dT), the elementary thrust on theelement.

In a similar manner it can be shown that the remainingforces make up a total force in a direction tangential to thedisc of revolution of the air-screw, and of amount

(dL. sin A 4- dD. cos A).This force comprises the drag or resistance exerted by the

element to circular motion, and is analogous to a frictionbrake applied to the rim of a revolving wheel. It tends toretard the motion.

It can be seen that the product of the above quantity andthe distance (x) of the element from the boss centre measuresthe torque of the element.

Thus, denoting the torque by (dM.), we havedM = x. (dL. sin A + dD. cos A),and we have already shown that the thrust on the element maybe expressed by dT = (dL. cos A-dD. sin A).

From these two equations and from the equation alreadyobtained for the lift on the aerofoil, we can solve all theproblems presented in the design of air-screws.We have

dT = dL. cos A dD. sin A,and we know that

cx drag dDtan i = =: Eft- = dL'


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18 AIPt-SCREWSwhence, substituting, we obtain

dTl = p.n2.Cy.b. (2.7r.x P. tan 7).\/P2 -f-4.7r2 ..?j2 . dx,and therefore

T = p.n2 . A (2.7r.a;-P. tan 7)VF+47T2^?. dx,Jr,

giving the total thrust on each blade of the air-screw.(r) denotes the length of each blade from the boss centre.(r ) denotes the length from the boss centre to the extremeportion of the blade where the streamlining of the sections

ceases. As a rule, in determining approximate values for theintegrals, we may take (?1 ) as equal to zero without appreciableerror.We have also

dM = x. (dL. sin A+d~D. cos A)= p.tf.Cy.l.x. (P+ 2.7r.x. tanand therefore

M = .?i2 . \C.b.x. P+ 2.7r.#. tanVP2+ 4.7T2 . 2 . dx.And since (M.2.7rji) is proportional to the B.H.P. requiredto turn each blade of the air-screw, we can at once determine

the necessary value of the blade width at each radius to satisfyany given set of conditions.We are now in a position to write down the efficiency ofthe whole blade, that is, of the air-screw.

The work done per revolution by the air-screw is theproduct of the total thrust exerted and the distance throughwhich it is exerted, that is, the quantity having the value ofY- and defined as the effective pitch of the air-screw.n

Let (N) denote the number of blades of the air-screw.Then we havework done^by air-screw per revolution = (N.T.P.),andwork done by motor per revolution in

turning the air-screw at (n) revs./sec. = (N.M.2.7T),


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FOKCES ACTING ON AN AIE-SCREW BLADE 19and therefore the efficiency of the whole air-screw is given by

N.T.P.V =

and /. T =

N.M.2.7T.

P. I l.Cy. (2.7T.-P. tan 7)VFT4Pr. 2 . dxJr5.7T. b.Cy.x.Jrn

.7r2.a2 . dx

This gives the general formula for the efficiency of any typeVof air-screw, for any specified value of .We may now endeavour to determine the values of the

pitch, corresponding to no thrust on the blades, to no torqueon the blades, and to no resultant average reaction on theblades.

The first of these quantities is called the ExperimentalMean Pitch, and we shall now endeavour to determine bycalculation the value of this quantity for any given set ofconditions. yThis value of the ratio is usually determined for anygiven air-screw experimentally in a wind-tunnel.The method to determine this theoretically is as follows.We have already shown that the expression for thethrust on each blade of any given type of air-screw maybe written as

T = p.n2 . \Cy.l. (2.7T.&-P. tan 7).Jr . dx,Ywhere P = .nNow if T = 0, then since (p.n*) is finite we have

fr= Cy.l. (2.7r.a;-P. tan 7). \/P2 +4.7ra.a;2. dx,JrC 2


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20 AIR-SCREWSfr __.e. 2.7T. I C.l).x. \/P2 -{-4.7r2 . 2 . dxr __.7T. I Cy.l).x. \/P2 -{-4.7r2 . 2 .Jr

= P. cy .t>.Jr tan 7. \P2+ 4.7r 2 . 2 . dor.And this may be solved graphically as follows.Take successive values of (P) from some value greater thanwhat would correspond to the effective pitch of the air-screw

upwards, and plot the two graphs 2.7r.%.6.#.\/P2 -{-4.7r2 .a;2 andP.Cy.6. tan 7. \/P2+ 4.7r2 .a32 against (x) between (?) and (r) foreach of the values of (P) taken. Take the areas of the twofigures thus obtained in each case. When the two areas in anycase are numerically equal, then the value of (P) taken for thiscase is the value of the experimental mean pitch of the air-screw.

It is of course apparent that the values of (cy) and (tan 7)will vary with each value of (P) taken.We can, however, determine their respective values in thefollowing manner.

Let us denote the successive arbitrary values of (P) takenby (P', P , P' , ...... ), and let us suppose that P' < P < P' .Cy. (2.7r.aj-P. tan 7).so that^R = p.?ia .6.cy . (P2 +4.7ra .aj2 ). sec 7. ^,

whenceR = p.n\ Ib.Cy. (P2+ 4.7r2.^2 ). sec 7. dx,

Jroand if this be now plotted against (x), we shall obtain the Load Grading Curve for the whole blade.

Since (tan 7) is usually nearly constant between (r) and


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24 AIE-SCEEWS(r ), and also of small amount, we may write (sec 7) equal tounity, and our expression for the total resultant force over thewhole blade then becomes

frR = p.w?. I b.Cy.Jr dx,and the curve obtained when integrating this expressiongraphically will not appreciably differ from the one obtained byinserting the rather troublesome (sec 7) under the integral sign.

Having then obtained the value of (R) for each blade of theair-screw, we may proceed to determine the value of (P) forwhich (B) has a zero value, in the same manner as for the caseof a zero value of the thrust on each blade.yIn thus estimating this value of it will be prudent toinsert the (sec 7) in our expression for (R).We have then

R = p.n2 . I b.Cy. (P2+ 4.7T2.#2 ). sec 7. dx,Jrand since (R) = 0, and (p.n 2) is finite, we obtain

(r= I hep (P2 -f 4.7r2 . 2). sec 7. dx.J^o/ c 2 c. 2Now (sec 7) = \/ I -f- 2, and -^ is always a positive quantity.Cy Cy

Hence (sec 7) is always a positive quantity, and therefore(cy) must be negative, when (R) is equal to zero, for at leastsome portion of the blade.


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25

CHAPTER III.BLADE SHAPE AND EFFICIENCY.

WE return now to a consideration of the formula alreadydeduced for the efficiency of any type of air-screw blade.It is obvious that, if the sections at every radius from the

boss centre along the blade were of such a form (if it weres*

possible) as to possess no drag, so that the value of would becxinfinite, the efficiency of the whole air-screw would be unity.That this is so may be seen by making (tan 7) equal to zeroin the expression already obtained for the efficiency of anytype of blade.We have

P. I Cy.b. (2.7r.x-l\ tan 7). \/W+^r\x\ dxn - J rV- rjr2.7r. cy.b.x. (P+ 2.TT.X. tail 7). \/P2 4- 4.7^.0?. dxJr

and now, making (tan 7) = 0, we getP. I Cy.b.2.7r x.

.7T.

. dx

Unfortunately, however, (tan 7) never has the value of zero,although the smaller its value the higher the efficiency andvice versa.

It is obvious, since (&) is a function of the radius (x), thatthe value of (77) will vary for different forms of blade outline,


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26 AIPv-SCREWS?and it therefore becomes necessary^to find some form whichwill give as large a value as possible to (?;) consistent withstructural considerations.

Let us consider the efficiency of any blade element atradius (x) from the boss centre.We have already shown that the nett thrust of the element,in a direction normal to the air-screw's disc of revolution, isgiven by dT = dL. cos A (1 -tan A. tan 7),and that the nett drag of the element, in a direction tangentialto the disc of revolution of the air-screw, may be written

c?R* = dlj. sin A-f-^D- cos A= dL. sin A (1 + cot A. tan 7),and therefore the work done by the element per revolution ofthe air-screw is

dT.P = P.


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BLADE SHAPE AND EFFICIENCY 27The maximum point of efficiency along the blade may be

found by putting

The point of maximum efficiency is found to be at a valueof (A) = 45-? that is, in the neighbourhood of 43.^ -*

Since tan A = [ ^ ), we can writej.TT.XJ, giving the value of (x) for the

2.7T. tanmaximum point of efficiency along the blade.

i' a' 3' 4'Radius. (x)

FlG. 8.

The point of maximum efficiency so obtained is only strictlytrue when (7) is supposecf to remain constant over the wholeblade, that is, for every radius.

ftBut in practice the values of - x- for the various blade sectionsCywill not be found to vary very greatly, and hence the point ofmaximum efficiency will not fall very far short of that givenby the formula.

It is obvious from the foregoing that the most efficient bladewould be one in which the whole of the blade surface wasconcentrated at the point of maximum efficiency. The bladewidth (&) would then become infinite and the length of theblade in the direction of the radius (x) would be infinitesimal.


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'28 AIB-SCKEWSAs, however, such a form of blade is impossible to constructwe are forced to adopt a compromise between the width of the

blade and the diameter of the air-screw.Let us first consider the point along the blade at which

the efficiency reaches a maximum value. We may regard (7)as a variable, and we will denote it by tan 1 ty(x), so thattan 7 = -ty(x) ; then we have

2.TT.X.7 7O

and for a maximum value of (%),-^ = 0, and ,^ is negative.ClX CLXSo that

and this expression gives the value* of (x) for which (rjx) is amaximum.In order to evaluate the above it is necessary to know theform of the function ty(x), i.e. (tan 7).If (tan 7) is constant over the whole blade, then ^r'(x)

vanishes, and the above expression reduces toP

.7r.tan(45-|)We thus obtain the original relation

A = 45 -for the angle at which the point of maximum efficiency occursalong the blade.Now we have not as yet prescribed any particular valueto (&), the width of the blade at radius (x).In the majority of air-screw blades the value of (Z>) variesfor different values of (x), i.e. for different radii along theblade.Now it is obvious that the blade should, from considerationsof overall efficiency, be wider at some radii than at others.Hence in designing our blade we may plot a curve of propor-


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BLADE SHAPE AND EFFICIENCY 29tional blade widths against radii (x\ so that the true or actualblade width for each value of (x) will be equal to the widthshown on the curve multiplied by some constant.We may thus write the blade width (&) as equal to theproduct of a constant and a function of (x), or

b = C.f(X).Now we have already assumed that (7) is a variable and

may have different values for different values of (x), and we/>

have denoted the ratio of by ^(x) y so thatCytan 7 = ty(x\ and therefore

7 = tan 1 Tfr(x), as already given./iBut if (7) and therefore is a variable or function of (x),Cy

so also is (cy\ the absolute lift coefficient of the section atradius (x). Hence we may also write this in the form

It has already been shown thatP. 2.7T

which when plotted against radii (x) will give a curve ofefficiency for each value of (x).Now suppose we regard the curve so obtained as our pro-portional blade width curve, so that at each point along theblade the actual blade width is proportional to the efficiencyat that point. We shall then ensure at any rate that our bladeis widest at its most efficient point. This will not of coursenecessarily give the most efficient form of blade outline, butthe efficiency of such a type of blade will be greater than thatof one possessing the same characteristics in other directionsbut having a uniform blade width throughout.

If we turn to the efficiency curve already given, we shallnotice the following.

(1) As (x) increases from zero up to about (J) the lengthof the blade, the efficiency rises very rapidly.


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30 AIR-SCREWS(2) As (x) increases, from the value corresponding to the

point of maximum efficiency, to its maximum value (i.e. thelength of the blade), the efficiency steadily decreases, althoughsomewhat slowly compared with its initial rise from zero upto its maximum value.

(3) Hence, after the point of maximum efficiency has beenreached, the blade becomes less and less efficient as we proceedto the tip. Hence the useful work done by the blade elementsdecreases towards the blade tip.

(4) This at once suggests the advisability of making theactual blade widths proportionally less at the outside radii thanthey would be if made exactly proportional to the efficiencycurve. The second curve in Fig. (9) would then be a more

efficient shape than the efficiency curve. The maximum ordinateis now further to the right than in the efficiency curve.

There are however other considerations bearing upon theproblem of the most efficient blade outline besides thosealready given.If we pursue still further the analogy with an aeroplanewing, it at once becomes apparent that there may be a limit tothe useful blade width possible in an air-screw.

In the case of an aeroplane having two or more superposedsurfaces (e.g. as in a biplane) it is well known that the lift ofthe lower wing is affected quite appreciably by the wash of thetop wing, and that the smaller the vertical distance betweenthe two surfaces the greater is the loss in lift on the lower wing.

This vertical distance between the wings of an aeroplane is


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BLADE SHAPE AND EFFICIENCY 31known as the gap, and the ratio between this distance andthe width of each wing in the line of flight is known asthe |f , ratio. This ratio has usually a value of betweenchord(8) and (1*2) and is often equal to unity in standard types ofaeroplanes.

The value of this vertical distance or gap in the case ofthe blades of an air-screw is equal to

P. cos Afor an air-screw having (N) blades.

(fl

FIG. 10.

It is the value of the vertical distance between any twoconsecutive helicoidal paths after ^th of a revolution and after2^ths of a revolution respectively.

This may be illustrated if we have recourse once again tothe paper cylinder.

Consider an air-screw having (4) blades. And draw on therectangular piece of paper (7) parallel lines at equal distancesapart, Fig. (10).


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32 AIK-SCREWSNow fold the paper into a cylinder having a depth as before

of (P), Fig. (11). Then the lines so drawn will represent thehelicoidal paths traced out by the same point on each of the (4)blades of the air-screw. We notice that the vertical distanceor gap between any two such consecutive paths is equal to

P. cos AIf then we make the width of the air-screw blade at any

FIG. 11.

radius (x) proportional to the vertical distance between anytwo consecutive helicoidal paths, we introduce an allowance forpossible blade interference, Fig. (12).

So that we haveP. cos A

whenceI a

P. cos AIT


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BLADE SHAPE AND EFFICIENCYCL

33


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34 AIR-SCEEWSAn air-screw having this type of blade outline has been

called by Mr. A. E. Low the Eational blade. We shallrefer to this type of blade by that name in what follows.There are other types of air-screw blades, and we shall nowconsider two of these further types.The first type considered is one which would at first sightseem to be the simplest possible type of blade, the blade ofuniform width. Here obviously (&) is independent of (x),since it is the same for all values of the radii.

FIG. 13.

Hence (&) = constant = c.f(x), whence /(,>;) = 1, and therefore(*)=().And (c) has always the value of the ratio

actual blade width at radius (x)scale blade width at radius (x) '

In this case we can take the scale blade width as beingequal to unity.

A blade having a constant chord width has been called byMr. A. E. Low the Normale blade. We shall use this namewhen referring to this type of air-screw blade.The second form of blade considered is one which from aconstructional point of view forms the limiting curve ofconstruction in which all the lamina pass through the boss.

If in any type of air-screw the blade widths at any pointcome outside this curve, it is not possible to construct such ablade without using offsets on the laminae.

Such a form of blade is shown in Fig. (13).


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BLADE SHAPE AND EFFICIENCY 35The form of blade outline is seen to be given by

b = B. cosec (A+ a),whencec.f(x) = B. cosec (A+ a),and therefore

c = B, and f(x)and therefore

PTcos a+ 2.Tr.x. sin a

and this expression does not greatly differ from

the one obtained by making (a) = 0. This, as will be seen later,greatly simplifies the subsequent evaluation of the quantitiescharacteristic of such a type of air-screw blade.

This form of blade may be termed the ConstructionalLimit type.We shall now apply the formulae already deduced forany type of air-screw blade to the four blade types alreadyconsidered, namely :

(1) The Efficiency Curve type of blade outline.(2) The Bational type of blade outline.(3) The Normale type of blade outline.(4) The Constructional Limit type of blade outline.

The three expressions for Thrust, Torque, and Efficiencyhave been shown to be given by

T = p.n2 . Icy.l. (2.ir,oj-P. tan 7). \/P2+ 4.7r2.a32. dx9Jrwhich may be written asT = p,n*.c. \f(x)4(x). (2.7r,9;-P.^(t,') ). \/FJr

giving the Thrust exerted by each blade of the air-screw.D 2


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36 AIR-SCREWSThe Torque on each blade is given by

M = p.n2 . \Cy.l.x. (P+ 2.1T.B. tan 7). \/P2+ 4.7r2 . 2 . dx,Jrwhich may be written asM = p.n\c. \f(x). 4>(x). x, (P+ 2.ir.x. ^(x) ). \/P2 +4.?r2^ dx.Jr

The total Efficiency of each blade, and therefore of the wholeair-screw, is given by

. \cy.b. (2.7T.E- P. tan 7). \/P2 + 4.7r2 ,-e2 . dx2.7T. \cy.b.x. (P+ 2.7r.. tan 7). \/PJ^o

which may be written asP. /(#). ^(a?). (2.ir.a; - P. ^0) )-2.7T. /(a;),

and we shall now apply these general results to the four typesof blade outline considered.

Efficiency Curve type of blade outline.// \ P. (2.-n-.a?-P.^(aQ)re J( ' ~ 2.7r.x. (P+ 2.7r.aj; ^(x) )'

so that the expression for the Thrust on each blade becomesr... . \/P2 -f-4.7r2^2 . (2.ir.a;-P. ^)) 2 . dx

2.7T


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BLADE SHAPE AND EFFICIENCY 37The Torque on each blade is given byM = ^?. fj(a>). (2.7r,*-P.J/7r >oThe total Efficiency of the whole blade is given by

rrI (f)(x). (2.7T.OJ-P. ^0))

2.7T. \(x). (2.7r.# P. tfrfa)). vP2+ 4.7r'2 .a52 . dx

Rational type of blade outline.

N.so that the expression for the Thrust on each blade becomes

2 Po^2 c f rT = - -' i '. \x. 6(x). (2.TT.X P. ty(x)\ dx.N Jr.The Torque on each blade of the air-screw is given by

J ro

And the total Efficiency of the whole blade is given byP. lx.(x). (2.TT.X- P. ^(x)).dx

2.7T.


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BLADE SHAPE AND EFFICIENCY 39 Constructional Limit type of blade outline.

\/PM-~4 7T2 X1Here /(a?) = p ' , so that the Thrust on each bladeis given by

The Torque on each blade is given byM = PAnd the total Efficiency of the whole blade is given by

' \ fr\

Jr.'As can be seen from the above three expressions for Thrust,

Torque, and Efficiency, the evaluation of each expression is asimple matter if we take (%) and ty(x) as being constants,that is if we assume the air-screw to have a uniform sectionthroughout. The approximation thus introduced does not affectthe accuracy of the expressions to such a material extent asmight be expected. The main difficulty in the prediction ofthe performance of an air-screw lies in the shape of the bladeoutline not being as a rule readily capable of being expressedby some simple function of the radius (x).

We shall now determine a few values for the expressionsalready deduced for Thrust, Torque, and Efficiency, in the fourcases already considered, when the blade section is assumed tobe uniform throughout. This makes cf>(x) and ty(x) constants,and we shall refer to these functions when considered as beingconstants by (cy) and (tan 7) respectively.We shall assume that (r ) = 0.

For the present we shall confine our attention to theexpressions for the efficiency of two only of the types of bladeconsidered.


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40 AIR-SCREWS Rational type of blade outline.We have

CrP. \x. (2.7T.X P. tan 7). dx

Jr = P. (4.7r.r-?).P. tan 7)97 = f~ - /. -n , Qirrr tan ^.7T. p.J'o

And now, if we write (Z) equal to the ratio of the effectiveppitch to the diameter of the air-screw, i.e. -r, we obtain= 2.Z. (2.7T-3. tan 7.Z)77 = TT. (4.Z+ 3.7T. tan 7) *

The substitution of (Z) for the ratio of pitch to diameter,p-y, puts the expression into a more convenient form.

Mr. A. K. Low has deduced practically the same expressionfor the efficiency of this type of blade, his formula being

_ '2.wV

~o -\irin

where his (w) is equal to ^ (as used here), and his (M') andLi(M' ), which are expressions for the average values of (tan 7)over the blade, are equal to (tan 7), supposed to be constantover the whole blade (as used here).

If we make these substitutions, the two expressions for theefficiency of this type of blade become identical.*We can introduce a still further simplification into theRational blade efficiency formula by taking the value of(tan 7) as 1/12. This is a very fair average value for theusual type of blade section employed in practice.

* See Paper on Air-screws read before the Aeronautical Societyin April, 1913.


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BLADE SHAPE AND EFFICIENCY 41We have then

_ 2.Z.(8w-Z)77 ~~ 7T. (16.Z+7T)'

This curve may now be plotted against values of (Z). Sucha curve is shown in Eig. (14).

FIG. 14.

Constructional Limit type of blade outline.We have

?? =P. f(2.7r^-P. tan 7) (P2+ 4.7r2.^2). dx

2.7T. x.(P+ 2.ir.x. tanJr5.P.(6.7r.P2.^-12.P3.

.7r'2^2 . dx.^2 . tan 7)

.P2.^. tan .7r3.rf3. tan 7)'


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42 AIE-SCEEWSand now, as before, making the substitution (Z) = -y, we obtainCv

5.Z. (G.T^Z^J^.Z^an 7-f3.7r 3 -4.77 2.Z. tan7)77 ~ TT. 30.Z3+ 20^rrZ2 . tanAnd if we make (tan 7) = 1/12, this reduces to

5.Z. (18.7r.Z2 -3.Z 3+ 9.7r 3 -7T 2.Z)=7T. (90.Z3+ 5.7T.Z-+ 45.77 2.Z+ 3.7T 3)'

and this curve may likewise be plotted against (Z). The curveof this function is shown in Fig. (14).It will be noticed from the two formulae deduced for both

the Eational and Constructional Limit efficiencies, thatneither expression contains (d), the diameter of the air-screw.

This is as might be expected, since (ij) is non-dimensional,and hence the expression for the same should only involveratios such as (Z).If we make (tan 7) equal constant, and therefore assume

that (cy) is constant over the blade, we may obtain quantitativeexpressions immediately evaluable for the case considered ofthe Eational blade shape. These are then as follows :

-d2

. (2.7r.d 3.P. tan 7)12\r M c-Tf-P-Y-^-Cy-n2 . (4.P+ 3.7T. tan 7. d)48

, . 2.C.P.7T.^ , . .and, since (b) = =, we may obtain the aboveN.\/P2 -f4.7r2 .a;2expressions for the Thrust and Torque in terms of the width ofthe air-screw blade at the tip, i.e. when (x) is equal to (?).We may denote this tip blade width by (br).

These expressions then become

_ 5r .9ia.p.^VFT7r' 5Q2 .' (2.7r.^-3.P. tan 7)12

.n'2.d2 . (4.P-f 3.?r. tan 7.


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BLADE SHAPE AND EFFICIENCY 43and the value of ( r), the width of the air-screw blade at thetip, is given by

C.P.TT.d

In deciding upon the best form of blade outline for theair-screw, -an ideal condition would appear to be obtained whenthe velocity in the slip stream is everywhere parallel to theaxis and uniform, that is when the thrust at any radius isproportional to that radius. The ideal thrust grading diagramwould thus be a straight line passing through the origin at theboss centre.We can investigate the form of blade outline necessary tosecure such a condition from the results already obtained inthe general case.

It has already been shown that for any form of bladeoutline whatever, the thrust at any radius is given by

(IT = c.M2 xx.2.7r.x-?.rx. z +4:.'Tr*.x2 . dxand the condition specified is that

dT = m.x.dx,whence

m.x = c.pMz.f(x)4(x).(2.Tr.x^f.and therefore __

c.p.n*.(x).(2.7r.x- P.^(a;)).v/P2 -f 4.7r 2.^'which gives the necessary form of blade outline to satisfy theseconditions.

Whence it is only necessary to plot a curve of

to obtain the required blade outline for any specified outsideconditions.


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44 AIR SCREWSIt will be noticed that when (#) has the value of

that is at a distance of about one inch from the boss centre, theblade outline curve f(x) becomes discontinuous, being of theform shown in Fig. (14A).Thus in estimating the characteristics of such a type of

FIG. 14A.

blade it is impossible to integrate from the value (r ) = 0, sincethe curve becomes discontinuous at the value -J^ of the2.7Tradius.

It becomes necessary therefore, when evaluating theexpressions characteristic of such a type of blade, to take for


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BLADE SHAPE AND EFFICIENCY 45the value of (r ) a value greater than _i^i?2. Since this value2.7Tis usually very small, it will be sufficient to take for (r ) avalue of anything from say (-^Q) to (J) of the length ofeach blade.

It is interesting to see how the efficiency of this type ofblade compares with those types already investigated.The efficiency of any type of blade is given by

( r ________P. f(x). $(x). (2.7r.-P. Tjr(#))VP 2 4-4.7ra .sc'2 . dx= Jr 'r 2 - v/ ;

.j.7r 2 .ic2 . dx

and taking the value of,, s _

we obtainTr

L ) I /yi //O'*L.It//. C6tXyJn

pand this becomes, after making the substitutions = (Z), andCb(r ) = -Q, and ^(x) = tan 7 = constant,=_3.772.Z

and if tan 7 = ,12

. log,


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46 AIR-SCREWSIt would appear that the curve of efficiency plotted against

values of (Z) for this type of blade does not greatly differ fromthose already given. The efficiency for the respective valuesof (Z) is slightly higher than that obtained for the Rational blade under similar conditions of blade section and anglesof attack.


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47

CHAPTER IV.BLADE SECTIONS, AND WORKING FORMUL/E.

IN air-screws manufactured for and used upon existing typesof air-craft it will almost invariably be found that the bladesection changes from the tip to the boss, and this is in fact anecessity, from considerations of strength due to the stresses inthe blades caused by the thrust exerted by the air-screw andthe centrifugal action due to the rotation of the whole air-screwabout its axis at the boss centre.

Aerodynamically the use of a varying form of section alongthe blade usually somewhat tends to decrease the efficiency ofthe air-screw as a whole, although this probably does notamount to much. That is to say that the efficiency of anytype of air-screw blade having a section varying from the bossto the

tipis less than that of a blade of similar outline but

having a uniform section throughout, provided that such asection is of such a form as to have a value of (tan 7) smallerthan the average value of (tan 7) on the other blade, andequal to the smallest value of (tan 7) on any section thereemployed.As a rule, for sections of similar shape, the thinner the

fil1 f* 1CTl f*^Ssection, i.e. the less the value of the - r -, ratio, the lowerthe value of (tan 7), and hence the greater the overall efficiencyof the whole blade employing such a section.

This only applies, however, between certain fairly well, ,. thickness ,.defined limits, for, after a certain value or the g -j ratio

is reached, the section becomes less and less efficient, that is hasa larger and larger value of (tan 7), until the limit is reached,


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48 AIE-SCREWSwhen the camber of the section altogether disappears and thesection ultimately becomes a flat plate for which the value ofthe YJ ratio is quite small, being of the order of (*7) : (1) at

.Dragthe most efficient angle of incidence.As a rule, in designing an air-screw to fulfil any givenoutside conditions, the forms of the blade sections are chosen

ChoYd Angle of Incidence (ctegrte*)FIG. 15.

so as to effect as far as possible a compromise betweenconsiderations of aerodynamical efficiency and the necessarystrength of the various portions of the blade.The aerodynamical considerations of overall blade efficiencyhave, however, already been dealt with, and the expression forthe efficiency of any type of blade shape put into a suitableform for purposes of air-screw design, so that we may proceed


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BLADE SECTIONS, AND WOEKING FORMULAE 49to select our sections along the blade without stopping toconsider whether it will be possible to express analyticallythe characteristics of the same in estimating the Thrust, Torque,and Efficiency of the whole blade, providing of course that theforms of the blade sections chosen are such that they conformto sections of which the characteristics are known from tests

Chord A ng/e of Incidence fdtyrreijFlG. 16.

carried out in a wind-channel for the same when consideredas aerofoils moving in a straight line.A somewhat typical series of such sections are given in the(< Technical Report of the Advisory Committee for Aeronauticsfor 1911-1912, and their characteristics are shown plotted inFigs. (15), (16), (17), (18), (19), (20), (21).

It will be noticed that the maximum value of the ^p-Drag


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50 AIR-SCKEWSratio, and hence the minimum value of (tan 7), occurs in all thesections at an angle of approximately (4) degrees.We may of course use any angle of attack we please foreach section along the blade, but it is advisable to use the anglecorresponding to the least value of (tan 7) fur each sectionconsidered, as this gives a better overall efficiency for the wholeblade, since the efficiency of any section is equal to

tan Atan (A +7)'

Chord Angle of Inc. i d e. neeFIG. 17.

and hence the smajler the value of (tan 7), and hence of (7),the greater the efficiency at that section.

We now proceed then to a consideration of the design ofany type of air-screw to fulfil any given specified set ofconditions.


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BLADE SECTIONS, AND WORKING FORMULAE 51It is usual to start with the following data, which are

supposed to be fixed from outside considerations.

(1) (V) The horizontal velocity of the air-craft.(2) (?i) The speed of revolution of the air-screw in

revs./second.

Chord Anyle of Incidence.FIG. 18.

(3) (d) The diameter of the air-screw. This isusually fixed from considerations ofground clearance, etc., and is usuallymade as large as the design of the air-craft will permit. This seems to befairly standard practice at present.Experimental research is required beforethis point can be definitely settled.

E 2


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52 AIR-SCEEWS(4)


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BLADE SECTIONS, AND WORKING FORMULA 53widths. Here experience and a study of the particular type ofair-craft to which the air-screw is to be fitted will guide us.

Portions of the air-craft coming within the air-screw's discof revolution must be taken into account when designing theblade shape.An air-screw fitted to a rotary motor having a large cowl,for instance, will not probably require so large a blade width

Chord Anyle of Inc i de nc tFIG. 20.

for best efficiency at the inside radii near the boss as wouldotherwise be advisable if the air-screw were working free fromany such obstructions in the air-flow.

In the theory of air-screw design as set forth in this bookno quantitative notice is taken of such stationary parts of theair-craft, and to do so would be in most cases an exceedinglydifficult problem.


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54 AIR-SCKEWSA study of the efficiency curve for the blade will also help

us in the designing of a suitable blade outline.Bearing then these considerations in mind we commence the

designing of our air-screw as follows :(1) Plot the efficiency curve

P. 2/

J Incidence (deyrees)FIG. 21.

We do not as yet know the value of the function ^r(,r),but for a first approximation we may take this asbeing constant and equal to say T^.Z

In slow running air-screws this curve of blade efficiencywill be found to be almost a parallel line with the(x) axis, except close up to the boss, where it runsdown to the origin very rapidly.


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BLADE SECTIONS, AND WORKING FORMULA 55In high speed air-screws the change in slope of the

efficiency curve will be more marked, and thisbrings us to a consideration of the best blade shapeunder these conditions.

As an example of a good blade outline for an air-screw, thefour-bladed air-screw as used on the Royal Aircraft Factoryaeroplanes may be cited. This screw revolves at a speed of900 r.p.m., which is if anything rather on the slow side.We may however take it as a fairly good general rule thatin designing an air-screw blade, the maximum ordinate on theblade outline curve, that is the point where the blade is widest,should be somewhat nearer the tip of the blade than the pointof maximum efficiency as given by the efficiency curve.*

FIG. 22.

It is not proposed here to discuss the theoretically best formfor the blade outline function f(x), as to do so would be a verydifficult matter when treated generally. Experience is a goodguide in choosing a suitable shape for the form of this function.

Taking then the blade outline as being something of the* Recent tests on aerofoils with elliptical-shaped ends have shown a

marked improvement, in ratio, over similar aerofoils with square-cut ends, provided that the section of blade in the former case iseverywhere geometrically similar. This indicates the superiority, fromefficiency point of view, of blades with tapered tips, apart from any otherconsiderations.

f A general treatment of the variation in (77) due to variation of /(#),and the determination of the form of f(x) giving to (77) a maximumvalue, would require the application of the Calculus of Variations, andis beyond the scope of the present work.


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BLADE SECTIONS, AND WOEKING FORMULAE 57quantitatively will approximate to the values of the character-istics shown on a smooth curve drawn between the variouspoints along the blade corresponding to the characteristics ofthe selected sections at those points.We shall then obtain a series of points for the values of(cy) and (tan 7) for the various radii chosen, and if smoothcurves be drawn through these points we shall obtain the twocurves denoted by


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58 AIE-SCEEWS

o *

'

/ vW


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BLADE SECTIONS, AND WOEKING FORMULAE 59available to turn the whole air-screw, where (N) denotes thenumber of blades.

Hence if we use Ib./ft./sec. units we get thatIN . ivi.-j.7T.n .^TK = J~ijooO

where (H) is the available B.H.P. after allowing for losses intransmission (if any transmission is employed).

We also have the value of (p) as being equal to ( 00238) inIb./ft./sec. units.So that we can at once write

= Hwhence

550. H(r2.7T.nS.N.p. X.f($).


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60 . AIE-SCEEWSnecessary to determine graphically the value of the definiteintegral

\X.f(x). $(X). (P-f2.7T.JE. ^)).V^ + ^7T\X\ dx,hand we may proceed to do this as follows.

Plot the graph of the functionx. f(x). $(x]

to some convenient scale against values of (x), the radius fromthe boss centre in feet. yWe already know the value of (P) ; it is ; and both (V)and (n) are known to start with, since they are assumed to befixed from outside considerations.

The value off(x) for any radius (x) is known, since it is thevalue of the ordinate on the scale blade width already drawn.The value off(x) is to be measured in feet, that is to say wemust take some convenient scale on the graph paper to representso many inches equal to one foot.The respective values of (x) and ^r(x) are determined atonce from a reference to the two curves already plotted of thesefunctions for any value of (x).

Having then taken a sufficient number of values of (x), andhaving determined the corresponding values of the function thegraph of which we are plotting, a smooth curve drawn throughthe points so obtained will give the curve required.Now draw two ordinates from the (x) axis at thepoints (r ) and (r) respectively until same cut the curve justplotted.Then the area of the enclosed figure so obtained, that is thefigure contained by the curve, the two extreme ordinates at (r )and (r), and the (x) axis, is the value of the definite integralrequired.The areas of closed figures of this kind are most easilyobtained by means of a planimeter.

It is to be noted here in this connection that the actual areaof the figure so obtained will be in, say, square inches, and


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BLADE SECTIONS, AND WOKKING FORMULAE 61hence that it must be multiplied by some constant in order tofind the real value of the definite integral.The value of this constant will depend, of course, upon thescale employed in the plotting of the graph.Thus, if the ordinate on the curve corresponding to anyvalue of (x) has a value obtained from the formula

T f(l-\ d\(t>\ (~P 4- 9 TT V ^(^\^J vV'V^V / \ -T^.TT.X.Yv*vof such and such an amount, this value will probably not beable to be represented on the graph paper used, since it may bevery much too large when taken to the same scale as that ofthe (x) axis, and hence the scale of heights or ordinates willprobably have to be made much smaller than the scale used forthe (x) axis.

Having then obtained the value of this area in the requireddimensions, we may determine the value of (c) at once.

It is given by550. H

/> ^ , ,

2.7r,n3.N.p. (area of figure obtained)'where (p) has the value of (-00238) as already given.We have then that the true or necessary blade width ateach radius (x) is the value of the scale blade width at thatradius multiplied by the value of the constant (c) alreadyfound.

OrI = cf(x).

The value of f(x) for any radius (x) will, of course, bemeasured off the curve of this function already drawn, andits corresponding real value found by reference to the scaleemployed in drawing the curve.

Thus, if 1 inch on the curve ordinate represents 1 foot asthe actual scale blade width, a value measured at any radiusof, say, ( ' 75) inch as the ordinate of the curve at that radiuswould represent an actual scale blade width of ( 75) foot, thatis (9) inches.And further, if the value of (c) was found to be, say, (1 2),then the true or actual blade width at this point would be


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62 AIR-SCREWSequal to (9 X 1*2) inches, that is (10*8) inches. And thiswould be the width of the blade at that radius to be used inthe construction of the air-screw.

We may obtain the value of the total efficiency of eachblade, and hence of the whole air-screw, in a similar manneras follows.

Plot the two curves4.7r^ (1)

andh4.^2 (2),

which is the one already plotted, and which therefore it isunnecessary to replot.

Take the area of the figure enclosed by (1), the extremeordinates at (r ) and (r), and the axis of (x).

The area enclosed by (2) has already been obtained from theevaluation of (c).

Divide the area enclosed by (1) by the area enclosed by (2),pand multiply the result by ^ . The answer will be theefficiency of the air-screw.

It will not, of course, be necessary to trouble about scaleconstants, etc., in determining the value of the efficiency of thewhole blade, as, provided that the two graphs (1) and (2) aredrawn to the same scale, it will only be necessary to dividetheir actual areas one by the other in whatever dimensionsthese two areas are obtained, provided, of course, that each areais measured in the same dimensions.In the determination of (c), and hence of the real bladewidth for each radius, it will usually be found that the trueblade widths for each point along the blade will be somewhatsmaller than those actually employed in practice on a similarform of air-screw. This is due to the fact that the calculatedTorque (M) is higher than the actual Torque found in practice,and hence that the necessary blade widths at each radius willhave to be larger than those given by the theory.

Of course no one for a moment supposes that the theory of


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BLADE SECTIONS, AND WORKING FORMULA 63air-screw design based on an analogy with aerofoils moving ina straight line is absolutely exact. It is merely a very usefultheory, the results of which conform very closely with thoseactually obtained by experiment.

It is difficult to estimate the amount of this differencebetween the calculated and actual Torques, without actualquantitative experimental results, but it would appear that theactual value of (c) is between -==- and ..- times the calculated

/ o 7ovalue as given by the theory.So that after having obtained the calculated value of (c), itwill be prudent to augment this value by say 33 %, that ismultiply the value of (c) obtained from the formula by -=- .7 o

It will be useful here to have some independent check uponthe working, so that any arithmetical slips in the evaluation ofquantities such as (c) may be as far as possible avoided.We may obtain a rather approximate check of this kindby reference to the Rational blade outline form alreadyconsidered.

If, in the expressions deduced for the characteristics of thisform of blade, we assume that both (cy) and (tan 7) are constantover the blade, and that (r ) is equal to zero, and further ifwe take the value of (tan 7) to be ^r, then we obtain the\2tfollowing expression for the necessary blade width constant (c).

52800. Hand since as already shown

c =where (br) denotes the tip blade width, we can obtain thenecessary value of (br) by substitution, thus

52800. HAnd this then provides a useful approximate check upon


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64 AIR-SCREWSthe previous work in the determination of the true blade widthfor each radius (#).

It will be noticed from this and previous expressions forblade width, that doubling the number of blades of an air-screwmerely halves the respective widths of same at correspondingradii.

This theory does not in fact take any notice of possibleimproved efficiency by the use of more than the customary twoblades for an air-screw.*

We may also obtain an approximate value for the efficiencyof the whole blade by means of the same Rational form ofblade outline.

This has already been shown to be given by2.Z. (2.7T - 3. tan 7. Z)TT. (4.Z+ 3.7T. tan 7) '

which reduces to the simpler form2.Z. (8.7T-Z)

77 == 7T. (7T+ 16.Z)when (tan 7) has the value of .

Now there will be, corresponding to some particular valueof (Z), a value of (rf) which will be a maximum, and this willtherefore give the theoretically best value of (Z) and thereforeof (n) to use for any given set of conditions, since the valuesof (V) and (d) will usually be fixed from outside considerations.We may obtain the value of this maximum efficiency andthe value of (Z) for which it is a maximum as follows.

The condition for a maximum value of (77) is^L-dZ ~ U 'and this gives

Zl = ~. [\/8 + 9. tan2~r7-3. tan 7],giving the requisite value of (Z) for a maximum value of (77).

* Except in so far as the greater the number of blades employed thehigher the aspect ratio of each, and hence the greater the efficiency ofthe air-screw due to increase of aspect ratio.


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BLADE SECTIONS, AND WORKING FORMULA 65This expression is seen to approximate to the value ^_f

as (tan 7) approaches zero.As a rule this value is in the neighbourhood of (2) andhence we may say that, for a near approximation, (77) has amaximum value when (Z) is equal to (2).P VNow (Z) = - = - and hence if (Y) and '(d) are fixed byCl> 1l'.(L

outside considerations, the value of (n) for which the efficiencyof the air-screw is a maximum is given by

Zl = ^d ~ 2 'whence(n L ) = y-j revs. /sec.

So that if(V) = 100 feet/sec.,(d) = 10 feet,

the speed at which the air-screw, or air-screws, should be run inorder to obtain the maximum efficiency would be (5) revs. /sec.,that is (300) revs./min.

This speed is of course abnormally slow in the light ofpresent-day practice, although the speed of revolution of theair-screws in some of the Wright aeroplanes is as slow as(450) revs. /ruin.A curve of efficiency for values of the ^i - ratio (Z)Diameterhas already been given for values of (Z) occurring in practice.

Mr. H. Bolas gives a formula for the efficiency of an air-screw of good shape * (see

Technical Report of the Advisory

* This formula is only an approximate one. It is 1 4- Ic. tan y. cot 6- ZZ 4- 7T./t. tan y'

and if tan y = , A' = *7, we getZ~ Z + '188'

which may then be plotted against (Z).


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66 AIK-SCKEWSCommittee for Aeronautics, 1911-12 ), and this has also beenplotted against values of (Z). The two curves of efficiency areshown in Fig. (14), and their close general resemblance will benoticed.

It would seem that at any rate to a first approximationthe efficiency of almost any modern type of air-screw may beobtained from the formula or graph given for the Eational shape.As a rule, in practice it is found that the actual recordedefficiencies of air-screws are higher than those given by thetheory.

Since (Z) may be taken as equal to (2) for a maximumvalue of (??), the maximum value of (77) will be obtained bysubstituting this value for (Z) in the efficiency formula. Weshall take (tan 7) as being equal to ^ as before, and thenwe have the value of (?;) as ('84). Hence the maximumoverall efficiency of a,n air-screw having a Eational blade

shape is (84%) when (tan 7) = ^ and (Z) = (2).Suppose that we wish to design an air-screw for an

aeroplane having a speed (Y) of (100) feet/sec., a rotationalspeed (11) of (20) revs./sec., a diameter (d) of (9) feet, andhaving a motor capable of developing an effective horse-power(H) of (100).Then

1} and hence (P) - Y- = 5 feet.( /Z' ) .. \J S Ji>

(d) = 9(H) = 100

Let the spacing of the sections along the blade be such asalready given with a uniform angle of attack of (4), andhence let the curves of $(#) and ^(x) be such as given inKg. (23).

Suppose that the form of /(#), the blade outline, is thatgiven in Fig. (22).


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BLADE SECTIONS, AND WOEKING FORMULAE 67Then we have to determine (1) The actual blade widths at

every point along theblade.

(2) The total efficiency of thewhole air-screw.We shall neglect the B.H.P. consumed in turning the

portion of the air-screw from the boss centre to the insideradius (r ), as by doing so we shall be on the safe side, sinceour blade widths will now tend to come out larger than if weallowed for the B.H.P. used near the boss. In any case theamount of this B.H.P. is very small.We proceed to determine the value of (c) in the mannergiven by plotting the graph of

x.f(x).


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68 AIR-SCREWScalculated and actual Torques, we obtain (3-07) as a moreexact value for this constant.We can now of course obtain the true blade widths at eachradius by multiplying their respective values as given on the

i

s

5colc o/ 5colr Blade WidthSome as Horizontal 5 c a I e .

scale blade width curve by (3 -07) when the air-screw has(2) blades.We can also check this value by applying the Rational blade constant formula and this then gives

52800. H^Tr.n*.p.cy.d?.N. (16.P + TTJ )V


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BLADE SECTIONS, AND WORKING FOEMUL^E 69= (-486) foot, which is the value of the width of the blade atthe tip, when (N) = (2). The value of (cy) has been taken tobe (*36). This value of the blade tip width is about whatmight be expected for this type of blade, where the bladeincreases in width towards the tip.We may now determine the value of (77), the total efficiencyof the whole blade. We plot the curve

and take the area of the figure enclosed by it, the two extremeradii at (r ) and (?'), and the (,/;) axis.The area of this curve as originally drawn is approximately(47 6) sq. ins. Eig. (24).The value of the total efficiency of the whole blade is thenequal to

47-6 \ / P \ /47-6\ / 5 \=

since (?) is equal to (5) feet.The efficiency of an air-screw of this type would thereforebe approximately 75 3 %.The blade sections of an air-screw at the inside radii nearthe boss have to be made thick from considerations of strength.We may, however, choose suitable shapes for the outside radiifrom sections which have a high value of the , ratio asDragaerofoils.A few typical examples of such suitable shapes are givenin Eigs. (25), (26), (27), (28).*

* See Technical Eeport of the Advisory Committee for Aeronautics,1912-13.


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AIR-SCKEWSDCS.CN /*e.fo).

FIG. 25.

CAorrf An,/* o/ tnci-Jtct.(d*yi)FIG. 26.


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BLADE SECTIONS, AND WORKING FORMULAE 71

FIG. 27.

FIG. 28.


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72 AIR-SCREWS

CHAPTEE V. LAYING OUT THE AIR-SCREW.

IN commencing to lay out the blade sections of an air-screw,it is first necessary to determine the chord angles of the bladeat several radii. Having done this-, the plan and elevation ofthe blade may be drawn out, consideration being given to thefact that as far as possible the two following conditions shouldbe satisfied :

(1) The centre of area of the sections should lie on theblade axis.

(2) The respective positions of the centres of pressure ofthe sections should be so arranged about the bladeaxis as to eliminate as far as possible all twist on theblade. The loading on the blade may be taken asbeing uniplanar.

FIG. 29.

These two conditions may be occasionally somewhatantagonistic.A symmetrical plan form is undesirable. A good plan formis shown in Fig. (29).


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LAYING OUT THE AIR-SCKEW 73We can obtain the true chord angles ((/>) for each radius

f the blade considered from the relation

(j) v = ax + tanwhere (P) has the value of as already denned.

The values of (ax) may vary along the blade, althoughusually it will be found that the values of the angles ofattack are approximately constant and in the neighbourhoodof 4 degrees.

If we draw, Fig (30), a vertical line to represent the valueyof and a horizontal line to represent (radii X 2.?r) in the sameunits, we may, by drawing in the various hypotenuses, obtainthe inclination of the helix paths for any element along theblade. And if these helix angles (A) be augmented by therespective angles of attack at these points, we shall obtainthe true chord angles for the various radii considered.

If, further, the widths of the blade at these radii be drawnin to scale along the chord angle lines, we may at once proceedto lay out the plan and elevation of the whole blade.*

Sections near the boss may be thickened up if necessary byadding a convex lower surface. In such sections the calculatedchord angles may have to be departed from. This is not ofgreat importance, although the actual chord angles of suchsections should not be less than their respective helix anglesat these radii.

Modern air-screw blades are built up of several separatelaminations of wood. French walnut is usually chosen asthe most suitable material from which to construct theair-screw.

The lamina? may be easily laid out when the chord angles* Strictly the blade widths and sections at each radius when obtained

should lie on cylindrical sections coaxial with the air-screw, and not onplane sections at right angles to a fixed arbitrary line in the blade. Thedifference, however, is small at all but the smallest radii, where it is ofleast importance.


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AIR-SCREWS


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LAYING OUT THE AIE-SCBEW


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76 AIE-SCEEWSat the several radii considered have been determined. Themethod is indicated in Fig. (31).The

plottingof the contours along the blade is obtainedfrom a consideration of the plan form of blade and the con-

struction of the laminae. ' A specimen contour plotting isshown in Fig. (. 32).In the laying out of the air-screw the various curves should

as far as possible be run into the boss with smooth curves, the

FIG. 31.

size of the boss being fixed from considerations of blade widthand type of air-screw mounting used.The thickening up of blade sections by means of a convexunder surface would not appear to affect their aerodynamicproperties to any great extent. Such thickening up maysometimes be necessary from considerations of strength.*

* The employment of a convex undersurface appears to slightlydecrease the value of (cy), and hence necessitates the utilisation of awider blade section. This is sometimes done when a stronger section isrequired. The LiftDrag ratio is only very slightly affected.


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77

CHAPTEE VI.STRESSES IN AIR-SCREAV BLADES.

Centrifugal Stresses.WHEN an air-screw is rotating about its axis at the boss centre,the various elements which go to make up each blade are forcedto follow a circular path. The forces necessary to make theseportions of the blade follow such circular paths are directedtowards the point about which the air-screw is rotating (i.e.the boss centre), and are of amount equal to

(weight of portion of blade considered) . (average velocityof portion considered) 2 divided by (g) . (averagedistance of portion from the boss centre).

Hence the reactionary forces with which the portions of theblade pull on any section considered are of the same amount,and constitute a stress in the material of which the blades ofthe air-screw are made.

Let Fig. (33) represent a blade of an air-screw, and let AA'be a section of same at a radius of (X) feet from the bosscentre. Then the centrifugal stress 011 the section AA' iscomposed of the

total pull exerted by the portion of theblade from A A' to the tip, divided by the area of the sectionat AA'.

Let (?) denote the overall length of each blade in feet.And consider the portion of the blade from AA' to the bladetip of length (r-X.) feet.

Consider the centrifugal pull on AA' due to a small elementof the blade cut off by the two radii (#) and (x + dv)respectively. Then (X) remains constant, while (./) varies fromthe value (X) to the value (r).


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78 AIR-SCREWSThen centrifugal pull on AA' due to the element considered

may be written ,/F _ (weight of element) (velocity of element)

2

$And let (w) = weight in Ibs. of one cubic foot of the material

of which the blades of the air-screw are made, and (iv) is thenassumed to remain constant for all values of (x).

FIG. 33.

And let (a) = area of element considered in sq. ins.And therefore

in sq. ft.

= area ^ sec^on f element considereda.dxAnd therefore volume of element = -Vr cu. ft.

And therefore weight of element = TJJ- Ibs.And circumferential velocity of element is equal to

(2.7r.x.n) ft./sec.


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STEESSES IN AIE-SCEEW BLADES 79So that we may write

giving the centrifugal pull on AA' due to the element con-sidered.

Hence the total pull on AA' is given byE = ' a.x.dx Ibs.

Whence the stress due to the centrifugal pull at AA' isriven by

a.x.dx Ibs./sq. in.,

where (c^) is equal to the area of the section at AA' insq. ins.Now () denotes the area in sq. ins. of the section atradius (#) from the boss centre, and consequently the valueof (a) may vary with (x). It is, however, a simple matter toEevaluate graphically the expression for , if necessary.We have thenStress due to centrifugal action at any section distance (X)Ffrom the boss centre = Ibs./sq. in.,where

P =T -I' = xWe may at once estimate the tensile stress, due to

centrifugal action, near the boss of the air-screw, if weO 7assume that

(1) The blade width is uniform, and(2) The blade section is constant, except near the boss.


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80 AIR-SCREWSThen we have fit once that

F 7r-.n\w.

'3Q.ff.a-i alx

since () is constant

And this formula holds good for any value of (X) providingthe initial assumptions (1) and (2) are satisfied.

Since near the boss (X) may be taken equal to zero, theformula becomes

F 7r 2.n2.w.a.r2 .. ,- = ^-r- - Ibs./sq. in., 7'2.g.a l

and this gives the value of the tensile' stress at or near the bossdue to the centrifugal pull of the whole blade.

An example will make the application of this result clearer.Let (n) = 20 revs./sec., (r) = 4 ft., (w) = 35 Ibs./cu. ft.,(a) = 7 sq. ins., and (i) = 16 sq. ins.

Then, taking the value of (?r2) as being equal to 10, we getF 10.400.35.7.16T ~T2.3216 ^ lbs./sq. in.

air screws intro du 00 ria crich

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